But at times, beams were portrayed to move extremely fast, thus giving 0 time for the target to move.

If star-ship phasers have a maximum effective range of 300 000 km, then it can also be interpreted that phasers move at the speed of light.
If same principles are applied to hand-held phasers, then targets shouldn't have any time to move in order to avoid the beam (unless the aim of the shooter is bad).

Quite simple.
I don't see the reason why there should be any trade-offs.
The beams were likely portrayed as moving slower on screen for special effects and giving a 'chance' to our heroes (and their opponents) to survive.
Otherwise, it would have been a very short story (under the same circumstances).

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